Scilab Function
Last update : 11/3/2005

linear_interpn - n dimensional linear interpolation

### Calling Sequence

  `vp = linear_interpn(xp1,xp2,..,xpn, x1, ..., xn, v [,out_mode])`

### Parameters

- `xp1, xp2, .., xpn`: real vectors (or matrices) of same size
- `x1 ,x2, ..., xn`: strictly increasing row vectors (with at least 2 components) defining the n dimensional interpolation grid
- `v`: vector (case n=1), matrix (case n=2) or hypermatrix (case n > 2) with the values of the underlying interpolated function at the grid points.
- `out_mode`: (optional) string defining the evaluation outside the grid (extrapolation)
- `vp`: vector or matrix of same size than `xp1, ..., xpn`

### Description

Given a n dimensional grid defined by the n vectors `x1 ,x2, ..., xn` and the values `v` of a function (says f) at the grid points :

          v(i1,i2,...,in) = f(x1(i1),x2(i2), ..., xn(in))

this function computes the linear interpolant of f from the grid (called s in the following) at the points which coordinates are defined by the vectors (or matrices) `xp1, xp2, ..., xpn`:

       vp(i) = s(xp1(i),xp2(i), ..., xpn(i))

       or vp(i,j) = s(xp1(i,j),xp2(i,j), ..., xpn(i,j)) in case the xpk are matrices

The `out_mode` parameter set the evaluation rule for extrapolation: if we note Pi=(xp1(i),xp2(i),...,xpn(i)) then `out_mode` defines the evaluation rule when:

       P(i) not in  [x1(1) x1($)] x [x2(1) x2($)] x ... x [xn(1) xn($)]

The different choices are:

  - "by_zero": an extrapolation by zero is done
  - "by_nan": extrapolation by Nan
  - "C0": the extrapolation is defined as follows:

             s(P) = s(proj(P)) where proj(P) is nearest point from P
                               located on the grid boundary.

  - "natural": the extrapolation is done by using the nearest n-linear patch from the point.
  - "periodic": `s` is extended by periodicity.

### Examples

    // example 1 : 1d linear interpolation
    x = linspace(0,2*%pi,11);
    y = sin(x);
    xx = linspace(-2*%pi,4*%pi,400)';
    yy = linear_interpn(xx, x, y, "periodic");
    xbasc()
    plot2d(xx,yy,style=2)
    plot2d(x,y,style=-9, strf="000")
    xtitle("linear interpolation of sin(x) with 11 interpolation points")

    // example 2 : bilinear interpolation
    n = 8;
    x = linspace(0,2*%pi,n); y = x;
    z = 2*sin(x')*sin(y);
    xx = linspace(0,2*%pi, 40);
    [xp,yp] = ndgrid(xx,xx);
    zp = linear_interpn(xp,yp, x, y, z);
    xbasc()
    plot3d(xx, xx, zp, flag=[2 6 4])
    [xg,yg] = ndgrid(x,x);
    param3d1(xg,yg, list(z,-9*ones(1,n)), flag=[0 0])
    xtitle("Bilinear interpolation of 2sin(x)sin(y)")
    legends("interpolation points",-9,1)
    xselect()

    // example 3 : bilinear interpolation and experimentation
    //             with all the outmode features
    nx = 20; ny = 30;
    x = linspace(0,1,nx);
    y = linspace(0,2, ny);
    [X,Y] = ndgrid(x,y);
    z = 0.4*cos(2*%pi*X).*cos(%pi*Y);
    nxp = 60 ; nyp = 120;
    xp = linspace(-0.5,1.5, nxp);
    yp = linspace(-0.5,2.5, nyp);
    [XP,YP] = ndgrid(xp,yp);
    zp1 = linear_interpn(XP, YP, x, y, z, "natural");
    zp2 = linear_interpn(XP, YP, x, y, z, "periodic");
    zp3 = linear_interpn(XP, YP, x, y, z, "C0");
    zp4 = linear_interpn(XP, YP, x, y, z, "by_zero");
    zp5 = linear_interpn(XP, YP, x, y, z, "by_nan");
    xbasc()
    subplot(2,3,1)
       plot3d(x, y, z, leg="x@y@z", flag = [2 4 4])
       xtitle("initial function 0.4 cos(2 pi x) cos(pi y)")
    subplot(2,3,2)
       plot3d(xp, yp, zp1, leg="x@y@z", flag = [2 4 4])
       xtitle("Natural")
    subplot(2,3,3)
       plot3d(xp, yp, zp2, leg="x@y@z", flag = [2 4 4])
       xtitle("Periodic")
    subplot(2,3,4)
       plot3d(xp, yp, zp3, leg="x@y@z", flag = [2 4 4])
       xtitle("C0")
    subplot(2,3,5)
       plot3d(xp, yp, zp4, leg="x@y@z", flag = [2 4 4])
       xtitle("by_zero")
    subplot(2,3,6)
       plot3d(xp, yp, zp5, leg="x@y@z", flag = [2 4 4])
       xtitle("by_nan")
    xselect()

    // example 4 : trilinear interpolation (see splin3d help
    //             page which have the same example with
    //             tricubic spline interpolation)
    getf("SCI/demos/interp/interp_demo.sci")
    func =  "v=(x-0.5).^2 + (y-0.5).^3 + (z-0.5).^2";
    deff("v=f(x,y,z)",func);
    n = 5;
    x = linspace(0,1,n); y=x; z=x;
    [X,Y,Z] = ndgrid(x,y,z);
    V = f(X,Y,Z);
    // compute (and display) the linear interpolant on some slices
    m = 41;
    dir = ["z="  "z="  "z="  "x="  "y="];
    val = [ 0.1   0.5   0.9   0.5   0.5];
    ebox = [0 1 0 1 0 1];

    XF=[]; YF=[]; ZF=[]; VF=[];
    for i = 1:length(val)
       [Xm,Xp,Ym,Yp,Zm,Zp] = slice_parallelepiped(dir(i), val(i), ebox, m, m, m);
       Vm = linear_interpn(Xm,Ym,Zm, x, y, z, V);
       [xf,yf,zf,vf] = nf3dq(Xm,Ym,Zm,Vm,1);
       XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
       Vp =  linear_interpn(Xp,Yp,Zp, x, y, z, V);
       [xf,yf,zf,vf] = nf3dq(Xp,Yp,Zp,Vp,1);
       XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf];
    end
    nb_col = 128;
    vmin = min(VF); vmax = max(VF);
    color = dsearch(VF,linspace(vmin,vmax,nb_col+1));
    xset("colormap",jetcolormap(nb_col));
    xbasc()
    xset("hidden3d",xget("background"))
    colorbar(vmin,vmax)
    plot3d(XF, YF, list(ZF,color), flag=[-1 6 4])
    xtitle("tri-linear interpolation of "+func)
    xselect()

### See Also

`interpln`, `splin`, `splin2d`, `splin3d`,

### Author

B. Pincon